Digital Lesson Graphs of Trigonometric Functions Properties of Sine and Cosine Functions The graphs of y = sin x and y = cos x have similar properties: 1. The domain is the set of real numbers. 2. The range is the set of y values such that 1 y 1. 3. The maximum value is 1 and the minimum value is 1. 4. The graph is a smooth curve. 5. Each function cycles through all the values of the range over an x-interval of 2 . 6. The cycle repeats itself indefinitely in both directions of the x-axis. Copyright by Houghton Mifflin Company, Inc. All rights reserved. 2 Graph of the Sine Function To sketch the graph of y = sin x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 3 x 0 2 2 sin x 0 2

1 0 -1 0 Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y = sin x y 3 2 2 1 2 3 2 2 5

2 x 1 Copyright by Houghton Mifflin Company, Inc. All rights reserved. 3 Graph of the Cosine Function To sketch the graph of y = cos x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 3 x 0 2 2 2 cos x 1 0 -1 0 1 Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A

single cycle is called a period. y = cos x y 3 2 2 1 2 3 2 2 5 2 x 1 Copyright by Houghton Mifflin Company, Inc. All rights reserved. 4 Example: Sketch the graph of y = 3 cos x on the interval [, 4].

Partition the interval [0, 2] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval. x 0 2 3 2 2 y = 3 cos x 3 0 -3 0 3 max x-int min x-int max

(0, 3) y (2 , 3) 2 1 1 2 3 2 3 4 x ( 3 , 0) 2 ( , 0) 2 ( , 3) Copyright by Houghton Mifflin Company, Inc. All rights reserved. 5 The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function. amplitude = |a| If |a| > 1, the amplitude stretches the graph vertically.

If 0 < |a| > 1, the amplitude shrinks the graph vertically. If a < 0, the graph is reflected in the x-axis. y 4 y = sin x y= 1 2 2 3 2 2 x sin x y = 4 sin x reflection of y = 4 sin x y = 2 sin x y = 4 sin x 4 Copyright by Houghton Mifflin Company, Inc. All rights reserved. 6

The period of a function is the x interval needed for the function to complete one cycle. For b 0, the period of y = a sin bx is 2 . b For b 0, the period of y = a cos bx is also 2 . b If 0 < b < 1, the graph of the function is stretched horizontally. y y sin 2 period: 2 period: y sin x x 2 If b > 1, the graph of the function is shrunk horizontally. y y cos x 1 y cos x period: 2 2 2 3 4 x period: 4 Copyright by Houghton Mifflin Company, Inc. All rights reserved.

7 Use basic trigonometric identities to graph y = f (x) Example 1: Sketch the graph of y = sin (x). The graph of y = sin (x) is the graph of y = sin x reflected in the x-axis. y = sin (x) y Use the identity sin (x) = sin x y = sin x x 2 Example 2: Sketch the graph of y = cos (x). The graph of y = cos (x) is identical to the graph of y = cos x. y Use the identity x cos (x) = cos x 2 y = cos (x) Copyright by Houghton Mifflin Company, Inc. All rights reserved. 8 Example: Sketch the graph of y = 2 sin (3x). Rewrite the function in the form y = a sin bx with b > 0 y = 2 sin (3x) = 2 sin 3x Use the identity sin ( x) = sin x: 2

2 period: amplitude: |a| = |2| = 2 = b 3 Calculate the five key points. x y = 2 sin 3x 0 6 3 2 2 3 0 2 0 2 0 y ( 2 , 2)

2 6 6 (0, 0) 2 3 2 3 2 5 6 x ( 3 , 0) 2 ( , 0) ( , -2) 3 6 Copyright by Houghton Mifflin Company, Inc. All rights reserved.

9 The Graph of y = Asin(Bx - C) The graph of y = A sin (Bx C) is obtained by horizontally shifting the graph of y = A sin Bx so that the starting point of the cycle is shifted from x = 0 to x = CC/BB. The number CC/BB is called the phase shift. y amplitude = | A| period = 2 /BB. y = A sin Bx Amplitude: | A| x Starting point: x = C/B Period: 2/B Copyright by Houghton Mifflin Company, Inc. All rights reserved. 10 Example Determine the amplitude, period, and phase shift of y = 2sin(3x-) Solution: Amplitude = |A| = 2 period = 2/B = 2B = 2/B = 23 phase shift = C/B = 2B = /B = 23 Copyright by Houghton Mifflin Company, Inc. All rights reserved. 11 Example cont. y = 2sin(3x- ) 3 2 1 -6 -5 -4 -3 -2 -1

1 2 3 4 5 6 -1 -2 -3 Copyright by Houghton Mifflin Company, Inc. All rights reserved. 12 a sin(bx c) d Amplitude Period: 2/B = 2 b Phase Shift: c/B = 2b Vertical Shift Copyright by Houghton Mifflin Company, Inc. All rights reserved. 13